Hodge and Teichm\"uller

We consider the derivative $D\pi$ of the projection $\pi$ from a stratum of Abelian or quadratic differentials to Teichm\"uller space. A closed one-form $\eta$ determines a relative cohomology class $[\eta]_\Sigma$, which is a tangent vector to the stratum. We give an integral formula for the pairing of of $D\pi([\eta]_\Sigma)$ with a cotangent vector to Teichm\"uller space (a quadratic differential). We derive from this a comparison between Hodge and Teichm\"uller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichm\"uller theory.